Development of Numerical Algorithm Based on a Modified Equation of Fluid Motion with Application to Turbomachinery Flow

Transcription

1 Development of Numerical Algorithm Based on a Modified Equation of Fluid Motion with Application to Turbomachinery Flow Von der Fakultät für Ingenieurwissenschaften, Abteilung Maschinenbau der Universität Duisburg-Essen zur Erlangung des akademischen Grades DOKTOR-INGENIEUR genehmigte Dissertation von Bo Wan aus Jiangsu, China Referent: Prof. Dr.-Ing. F.-K. Benra Korreferent: Prof. Dr. S. H. Sohrab Tag der mündlichen Prüfung: 03. September 2012

2 Abstract On the basis of the scale-invariant theory of statistical mechanics, Sohrab introduced a linear equation termed the modified equation of fluid motion. Preliminary investigations have shown that this modified equation can be extended to solve flow problems. Analytical solutions of basic flow problems were derived using this equation. In all cases the match between estimated and experimental data was good. These results stimulated further applications of this modified equation in the development of a CFD code to obtain numerical solutions of turbomachinery flow problems. In the present work, a novel numerical algorithm based on the aforementioned modified equation has been developed to solve turbomachinery flow problems. In order to avoid dealing with more technical conditions on the scale invariant form of the energy equation, this investigation is restricted to incompressible flow. On the basis of the work done by Sohrab, the derivation process of the modified equation for incompressible flow is presented with more emphasis on its linear property as compared to the Navier Stokes equation for incompressible flow. Furthermore, a detailed analysis of the present discretisation technique for the modified equation is performed. As compared with the Navier Stokes equation, the numerical errors resulted from the discretisation of the modified equation, including the truncation and discretisation errors are discussed as well as the stability conditions. On the basis of the analysis above, a novel numerical solver is established by using the open source code OpenFOAM. A corresponding iteration technology is determined for the solver to dedicate reliable and efficient solutions to the algebraic linear equations derived from the FVM discretisation of the modified equation. In addition, a dynamic mesh solver developed on the basis of the ALE method is also built up to resolve the transient flow problems. Finally, the developed solver is applied to solve the modified equation for several flow models including the fundamental boundary layer problems, flow around an airfoil, turbulent secondary flow in a curved duct and cascades, and rotor stator interactions in radial pumps. The results are evaluated by comparing them with those obtained by other methods, including the numerical results from a Navier Stokes solver and measured data. For all investigated cases the computational effort and the accuracy of the solver are emphasized. The comparisons indicate that the developed solver, which is based on the modified equation of fluid motion, requires less than half the computation time of the Navier Stokes solver, and it produces physically reasonable results validated by measured data. The encouraging result demonstrates the scientific credibility of the developed solver for application to turbomachinery flows. Further, if it is extended to design processes in the turbomachinery industry, significant improvements in the design cycle cost can be reached. II

3 Acknowledgments I would like to express my gratitude to all those who helped me during my study in Germany. My sincere gratitude goes first and foremost to Prof. Dr.-Ing. F.-K. Benra, my supervisor, for his inspirational guidance and constant encouragement. He offered me the opportunity to start this interesting topic, and gave me great help by providing me with necessary materials, advice of great value and inspiration of new ideas during my research. I am extremely grateful for his support in the accomplishment of my study in Germany. High tribute shall be paid to Prof. Siavash H. Sohrab, Northwestern University, IL, who introduced the scale-invariant form of the equation of motion, for his instructive theoretical support to my research and the review of my dissertation. I am also deeply indebted to all my colleagues and student assistants in the chair for Turbomachinery, University of Duisburg-Essen. Special thanks should go to Dr.-Ing. H. J. Dohmen who has given me valuable suggestions and patient guidance. Last but not the least, my gratitude also extends to my family who has been assisting, supporting and caring for me throughout my life. Bo Wan Duisburg, October 2012 III

4 Contents Contents 1 INTRODUCTION STATE OF THE ART OF THE MODIFIED EQUATION OF FLUID MOTION AND ITS APPLICATIONS MOTIVATIONS PRESENT CONTRIBUTIONS THESIS STRUCTURE MODIFIED EQUATION OF FLUID MOTION SCALE INVARIANT MODEL OF STATISTICAL MECHANICS CONVECTIVE, DIFFUSION AND LOCAL VELOCITIES MODIFIED EQUATION OF FLUID MOTION FOR INCOMPRESSIBLE FLOW Derivation of the Modified Equation of Fluid Motion Relation between the Modified Equation and the Navier Stokes Equation REYNOLDS AVERAGED MODIFIED EQUATION OF FLUID MOTION NUMERICAL ERROR TAYLOR SERIES EXPANSION TRUNCATION ERROR DISCRETISATION ERROR NUMERICAL DIFFUSION DEVELOPED SOLVER D DISCRETISATION CD Discretisation UD Discretisation SOLVER ALGORITHM & ITERATION Navier Stokes Solver Sohrab Solver DYNAMIC MESH SOLVER Navier Stokes Solver IV

5 Contents Sohrab Solver APPLICATIONS OF THE SOHRAB SOLVER BOUNDARY LAYER FLOW OVER A FLAT PLATE Laminar Flow Analytical Solution Numerical Solution Results & Comparisons Numerical Error Comparisons Turbulent Flow FLOW AROUND AN AIRFOIL Model Laminar Flow Numerical Simulation Setup Results & Comparisons Turbulent Flow SECONDARY FLOW IN A CURVED DUCT Numerical Simulation Setup Results & Comparisons SECONDARY FLOW IN CASCADES D Cascade Numerical Simulation Setup Results & Comparisons D Cascade Numerical Simulation Setup Results & Comparisons ROTOR STATOR INTERACTION ERCOFTAC Radial Pump Geometry of the Pump Stage Numerical Simulation Setup Results & Comparisons Symmetrical Radial Pump V

6 Contents Geometry of the Pump Stage Numerical Simulation Setup Results & Comparisons CONCLUSIONS OUTLOOK BIBLIOGRAPHY VI

7 List of Figures List of Figures: Fig. 2.1 A scale invariant view of statistical mechanics from cosmic to tachyonic scales [3]... 6 Fig. 2.2 Hierarchy of statistical fields for equilibrium eddy, cluster, and molecular dynamic scales and the associated non equilibrium fields [1]... 7 Fig. 2.3 Flow distribution near the source... 9 Fig. 2.4 Flow chart of the convective velocity calculation process Fig D mesh Fig. 3.2 Control volume V [29] Fig. 3.3 Variation of the volume flow rate Fig D FVM mesh Fig. 4.2 Flow chart of the calculation process of the Navier Stokes solver in OpenFOAM Fig. 4.3 Flow chart of the calculation process of the Sohrab solver in OpenFOAM Fig. 4.4 Flow chart of the Navier Stokes solver for dynamic mesh Fig. 4.5 Flow chart of the Sohrab solver for dynamic mesh Fig. 5.1 Laminar boundary layer flow over a flat plate Fig. 5.2 Assumption of the velocity inside the boundary layer Fig. 5.3 Numerical model of the laminar boundary layer flow over a flat plate Fig. 5.4 Grid for the laminar flow over a flat plate Fig. 5.5 Calculated velocity profiles inside the laminar boundary layer Fig. 5.6 Comparison of the velocity profile inside the laminar boundary layer between the numerical and analytical solutions of the modified equation of fluid motion Fig. 5.7 Comparison of the velocity profile inside the laminar boundary layer between the numerical solutions derived from different convective velocities Fig. 5.8 Comparison of the computation time taken by both solvers VII

8 List of Figures Fig. 5.9 Comparison of the velocity profile inside the laminar boundary layer between the experiment and numerical solutions Fig Comparison of the magnitude of the convective velocity and boundary layer velocities Fig Comparison of the numerical results influenced by the numerical error Fig Numerical model of the turbulent boundary layer flow over a flat plate Fig Grid for the turbulent flow over a flat plate Fig Comparison of the velocity profile inside the turbulent boundary layer Fig Geometry profile of the airfoil RAF Fig Complete grid region for the airfoil at α = Fig Detailed grid for the airfoil at four angles of attack Fig Potential flow over the airfoil Fig Comparison of the drag coefficient for diverse Reynolds numbers.. 60 Fig Comparison of the drag coefficient for different angles of attack Fig Comparison of the drag coefficient Fig Flow field near the trailing edge calculated by the Navier Stokes solver Fig Flow field near the trailing edge calculated by the Sohrab solver Fig Comparison of the pressure over the airfoil Fig Comparison of the velocity gradients inside the turbulent boundary layer Fig Geometry of the curved duct Fig Mesh effect on velocity profiles Fig Section of the grid of the curved duct Fig Comparison of the convergence process of the velocity Fig Comparison of computation time taken by both solvers Fig Vector plots of the velocity at various cross sections Fig Sketch of the cross section of the curved duct VIII

9 List of Figures Fig Comparison of the crosswise velocity profiles inside the bend at θ = Fig Comparison of the streamwise velocity profiles inside the bend at θ = Fig Comparison of the crosswise velocity profiles downstream the bend Fig Comparison of the streamwise velocity profiles downstream the bend Fig D model of the seven-blade 2D cascade Fig Part grid region of 2D cascade Fig Grids near leading edge and trailing edge Fig Comparison of the convergence processes of the lift coefficient Fig Comparison of the averaged computation time taken by both solvers Fig Comparison of the pressure coefficient Fig Comparison of the lift coefficients Fig D model of the six-blade annular cascade Fig Part grid of annular cascade Fig Grids near the leading and trailing edges Fig Comparison of the computation time taken by both solvers Fig Comparison of the pressure contours at pressure side of the blade Fig Comparison of the pressure contours at suction side of the blade Fig Position of the polyline for the wake plot Fig Comparison of the wake characteristics Fig ERCOFTAC pump model [55] Fig D mesh of the complete pump stage Fig Grids near the both leading and trailing edges of the impeller blade 91 Fig Grids near the both leading and trailing edges of the diffuser vane.. 92 Fig Comparison of computation time taken by both solvers Fig Comparison of the static pressure distribution IX

10 List of Figures Fig Comparison of the static pressure distribution in the enlarged blade tip region Fig Comparison of the absolute velocity fields at φ = Fig Comparison of the absolute velocity fields at φ = Fig Comparison of the absolute velocity fields at φ = Fig Comparison of the absolute velocity fields at φ = Fig Velocity triangle at the impeller outlet Fig Circumferential and radial components of the velocities Fig Comparison of relative velocity components in radial gap region at φ = Fig Comparison of relative velocity components in radial gap region at φ = Fig Comparison of relative velocity components in radial gap region at φ = Fig Comparison of relative velocity components in radial gap region at φ = Fig Pump view [54] Fig D mesh of the complete pump stage Fig Grids near the both leading and trailing edges of the impeller blade. 103 Fig Grids near the both leading and trailing edges of the diffuser vane 104 Fig Comparison of computation time taken by both solvers Fig Comparison of the absolute velocity fields at φ = Fig Comparison of the absolute velocity fields at φ = Fig Comparison of the absolute velocity fields at φ = Fig Comparison of relative velocity components in radial gap region Fig Comparison of relative velocity components in radial gap region at φ = Fig Comparison of relative velocity components in radial gap region at φ = X

11 Lists of Tables List of Tables: Tab. 5.1 Boundary conditions setup for the laminar boundary layer flow Tab. 5.2 Numerical schemes setup for the laminar boundary layer flow Tab. 5.3 Boundary conditions of the test cases Tab. 5.4 Boundary conditions setup for turbulent boundary layer flow Tab. 5.5 Boundary conditions setup for the laminar flow around an airfoil Tab. 5.6 Numerical schemes setup for the laminar flow around an airfoil Tab. 5.7 Boundary conditions setup for the turbulent flow around an airfoil Tab. 5.8 Numerical schemes setup for the turbulent flow around an airfoil Tab. 5.9 Boundary conditions setup for the flow in a curved duct Tab Numerical schemes setup for the flow in a curved duct Tab Boundary conditions setup for the flow in a 2D cascade Tab Boundary conditions setup for the flow in an annular cascade Tab Specifications of the pump stage [57], [58] Tab Boundary conditions setup for the flow in a radial pump Tab Numerical schemes setup for the flow in a radial pump Tab Specifications of the pump stage [54] Tab Boundary conditions setup for the flow in a symmetric radial pump XI

12 Nomenclature Nomenclature Variables A Area a, a Velocity coefficients C d C l Co C p C c Drag coefficient Lift coefficient Courant number Pressure coefficient Absolute velocity Chord D* Dimensionless distance (distance/boundary layer length) D Thermal viscosity e Error F d F l F F f x f H Kn k l m n O Pe Drag force Lift force Force Flux Distance rate Distribution function Delivery head, duct height, matrix Knudsen number Boltzmann constant, turbulent kinetic energy Span Mass Number, rotating speed Truncation error Peclet number p Kinematic pressure p* Dimensionless pressure (p/p max ) p Pressure XII

13 Nomenclature Q R Re r r T t S φ S s U b U u Flow rate Radius Reynolds number Position Distance/axis of rotation Temperature Time Source Distance, surfaces Surface Moving mesh velocity Circumferential velocity, element velocity, local velocity Atom velocity u τ Friction velocity ( τ / ρ ) V v gh, v g, v h Volume Diffusion velocities v* Dimensionless velocity (v/v in ) v W w Peculiar velocity Relative velocity x, y, z Coordinates Convective velocity, system velocity y+ Dimensionless distance between the first grid node and wall Z Number XIII

14 Nomenclature Greek symbols α β δ ε φ Γ γ ϕ μ ν ρ τ σ Ω ω ξ Ψ ψ Angle of attack Scale Difference, distance Energy dissipation rate Flux, transport variable Diffusion Blending factor Impeller postion Viscosity Kinematic viscosity Density Shear stress Stress tensor Local velocity / convective velocity Angular speed Similarity variable Arbitrary variabel Invariant function of velocity XIV

15 Nomenclature Abbreviations 1D One-dimensional 2D Two-dimensional 3D Three-dimensional ALE Arbitrary Lagrangian-Eulerian B. C. Boundary Condition BD Blend Differencing CD Central Differencing CFD Computational Fluid Dynamics CV Control Volume DGCL Discrete Geometric Conservation Law E East FVM Finite-volume Method GGI General Grid Interface HOT High Order Terms LCD Laminar Cluster Dynamics LED Laminar Eddy Dynamics LFD Laminar Fluid element Dynamics LHS Left-hand Side NVD Normalized Variable Diagram N-S Navier-Stokes OpenFOAM Open Field Operation and Manipulation PS Pressure Side RAF Royal Air Force RHS Right-hand Side SS Suction Side SST Shear Stress Transport TVD Total Variation Diminishing UD Upwind Differencing W West XV

16 Nomenclature Superscripts Fluctuating - Mean value n Time step XVI

17 Nomenclature Subscripts 1 Impeller inlet 2 Impeller outlet 3 Diffuser inlet 4 Diffuser outlet β a cw d E,N,P,W e,w free f in i J M m N-S Num out r sw t u w Scale Axial component Crosswise Diffuser Cell centre Node Freestream Cell face centre Inlet, inside Impeller Scale Modified equation, Sohrab solver Mean value Navier-Stokes solver Numerical Outside Radial component Streamwise Tangential component, turbulence Circumferential component Wall XVII

18

19 Chapter 1 Introduction 1 Introduction Computational fluid dynamics (CFD) methodology is widely used by engineers and designers in a broad range of industries to gain deeper insight into the effects of fluid dynamics on design processes. The fundamental governing equations for solving most of the CFD problems are the Navier Stokes equations. Under most conditions, the equations are second order, non homogenous, non linear and partial differential equations. The primitive variable formulations of these equations result in the derivation of a large system of non linear algebraic equations after the finite volume method (FVM) discretisation is carried out. However, these equations increase the associated computational cost and impede the development of numerical solutions obtained using the Navier Stokes equations. Hence, it is assumed that engineers would be interested in another equation which does not have non linear properties. The scale invariant theory of statistical mechanics has been developed in recent years. On the basis of a scale invariant model, Sohrab [1] introduced a linear equation termed the scale invariant form of the equation of motion or the modified equation of fluid motion. Preliminary investigations have shown that this modified equation can be extended to solve incompressible flow problems. Several basic flow models were analytically developed using this equation. Consequently, satisfactory correlation between the estimated and experimental data was reached. This result then stimulated further application of this modified equation in the development of CFD codes to obtain more numerical solutions of turbomachinery flows. 1.1 State of the Art of the Modified Equation of Fluid Motion and its Applications In 1992, a canonical grand unified theory of fields was introduced by Sohrab [2] to give the first impression of the application of the scale invariant statistical theory of fields to entire spectrum of scales from galactodynamics to subquantum tachyon dynamics. The appearance of this new theory offers another understanding for natural phenomena in diverse branches of modern sciences. On the basis of the scale invariant theory of fields, in 1999, Sohrab [3] presented a scale invariant model of statistical mechanics for equilibrium and non equilibrium scales. With the help of this model, the invariant definition of the local, convective, and diffusion velocities were introduced. Furthermore the scale invariant form of the equation of motion which is also called as the modified equation of fluid motion was derived from the scale invariant form of the Boltzmann equation [1]. In mathematics, this modified equation of fluid motion is nearly identical to the famous Navier Stokes 1

20 State of the Art of the Modified Equation of Fluid Motion and its Applications equation for incompressible flow. In comparison, the former replaces the local velocity in the convection acceleration component using the convective velocity which is different from the local velocity. Therefore, the non linear property is eliminated in the modified equation. This important feature uncovers the linear advantage of the modified equation of fluid motion in CFD applications as compared with the Navier Stokes equation. In order to prove the ability of the modified equation of fluid motion, several flow models have been analytically developed using this equation. In 1999, Sohrab [4] presented the first analytical solutions of the modified equation of fluid motion for two fundamental flow problems which were the laminar flow over a flat plate and through a circular pipe. For the flat plate problem, the resultant velocity profile inside the boundary layer, expressed by an error function, was compared with the classical theory of Blasius [5]. For the circular pipe, the resultant velocity distribution was compared with the parabolic profile of the classical theory of Hagen and Poiseuille [6]. It was found that both comparisons delivered satisfactory agreement between the modified and classical solutions. In 2006, the modified equation was solved for the laminar flow around a rigid cylinder [7]. Using an ultra low Reynolds number, the analytical solutions for the stream line around the cylinder and the thin boundary layer were compared with the numerical solutions obtained by the Navier Stokes equations. The scale invariant statistical theory also offers another understanding for the turbulence transition. As discussed by Sohrab [8], the potential energy is responsible to preserve the stability of the laminar flow particles. The increase in the particle velocity decreases the potential energy. When the critical velocity is reached, the potential energy vanishes, and the entire laminar field makes the transition to a fully developed turbulence field. This recovers the strong relation between the laminar and turbulence fields for the fluid element, and offers the possibility to solve the modified equation of fluid motion for the turbulent flow scale. An application of this theory can also be found in literatures [8, 9]. The modified equation was solved for the turbulent flow over a flat plate analytically. It is found that both flow fields in the outer laminar layer as well as the inner sub-layer are actually turbulent flow fields that are being convected by a global velocity. However, the associated analytical method was too complex to be directly used in the numerical application. Another treatment for solving the modified equation for turbulence problems is required. In 2008, for the laminar boundary layer flow problem Sohrab [10] presented a more comprehensive comparison between the predicted velocity profile of the modified equation of fluid motion and several experimental results, including Burgers and Zijnen [11 13], Hansen [14], Dhawan [15], Büttner and Czarske [16] and Nikuradse [17]. Additionally, the predicted velocity profile was compared with the numerical 2

21 Chapter 1 Introduction solution obtained by the Navier Stokes equations. It was shown that except some of the Nikuradse s data, all experimental data agreed with the prediction of the modified equation however systematically deviate from that of the classical Navier Stokes equations. These results provide further supports to the CFD application of the modified equation of fluid motion in accuracy. 1.2 Motivations All the investigations above contribute the understanding of the modified equation of fluid motion in mathematics, and give the proof for the applications of the modified equation with some fundamental flow models analytically. However these analytical methods cannot go further to solve the modified equation for complicated boundary conditions. This limits the scientific investigations of the modified equation in CFD applications. In order to expose the potential advantage of the modified equation of fluid motion for turbomachinery applications, the present research goes deeper insight into the modified equation using the numerical method of approach. In this context, the open source CFD toolbox OpenFOAM (Open Field Operation and Manipulation) is employed to establish a numerical solver using the modified equation. This developed solver must support the required mesh topologies and be available to use the FVM to discretize the modified equation of fluid motion. One key attribute of this solver is that with all supported flow models, it must produce accurate solution with fast and reliable convergence. Furthermore, as compared to the similar CFD solvers dealing with the Navier Stokes equations, it must deliver more efficient solutions for the same CFD problems, in terms of its linear property. 1.3 Present Contributions In this work, the investigation is restricted to incompressible flow to avoid dealing with more technical conditions on the scale invariant form of the energy equation. Referring to the motivations described above, this research makes the following contributions to the numerical solution of the modified equation of the fluid motion: (1) On the basis of the work done by Sohrab, the derivation of the modified equation of the fluid motion for incompressible flow is presented with more emphasis on its linear property as compared to the Navier Stokes equation for incompressible flow. 3

22 Present Contributions (2) In order to apply the finite difference method to discretize the modified equation, a detailed analysis of the present discretisation technique for the modified equation is performed. As compared with the Navier Stokes equation, the numerical errors resulted from the discretisation of the modified equation, including the truncation and discretisation errors are discussed as well as the stability conditions. (3) On the basis of the analysis above, a novel numerical solver is established by using the open source FVM code OpenFOAM. A corresponding iteration technology is determined for the solver to dedicate reliable and efficient solutions to the algebraic linear equations derived from the FVM discretisation of the modified equation. In addition, a dynamic mesh solver developed on the basis of the arbitrary Lagrangian Eulerian (ALE) method is also built up to resolve the transient flow problems. (4) Finally, the developed solver is applied to solve the modified equation for several flow models including the fundamental boundary layer problems, flow around an airfoil, turbulent secondary flow in a curved duct and cascades, and rotor stator interactions in radial pumps. As compared with the Navier Stokes solver, the mesh resolution, numerical scheme and computation time required for the developed solver are investigated. In order to examine the accuracy of the solver, the obtained results are validated by comparing them with the former experimental and analytical data reported in the literatures. 1.4 Thesis Structure The thesis is organized as follows. In Chapter 2, the scale invariant statistical theory related to the fluid dynamics scales is briefly introduced and a detailed description of the modified equation of fluid motion for incompressible flow is presented. In Chapter 3, the numerical errors caused by the space discretisation of the modified equation of fluid motion are discussed to give an access for the solver establishing by using the FVM. In Chapter 4, a developed solver for the modified equation of fluid motion is introduced in the OpenFOAM environment. In Chapter 5, the simulations of several flow models are discussed to explain the applications of the developed solver. In Chapter 6, the present research is concluded. In Chapter 7, suggestions for future investigations are proposed. 4

23 Chapter 2 Modified Equation of Fluid Motion 2 Modified Equation of Fluid Motion In 1999, Sohrab [3] gave an introduction to the physical and mathematical foundation of a scale invariant statistical theory of fields. The construction of the associated mathematical models would be greatly helpful to some fundamental conceptual difficulties which are encountered in engineering applications. In this chapter, on the basis of the scale invariant theory, the derivation of the modified equation of fluid motion will be discussed in detail. Furthermore, the linear advantage of this equation will be emphasized, as compared with the classical Navier Stokes equations. In addition, the time averaged expression of the modified equation for turbulent flow will also be delivered. 2.1 Scale invariant Model of Statistical Mechanics A scale invariant model was established for both equilibrium statistical mechanics, which is governed by the Gibbs function, and non equilibrium field, which is based on the Liouville, Boltzmann and Maxwell methods. As shown in Fig. 2.1, the broad scales from the exceedingly large scale of cosmic to the minute scale of quantum optics are included in the scale invariant model [3]. For equilibrium dynamics, the scales are equilibrium galacto, planetary, hydrosystem, fluid element, eddy, cluster, molecular, atomic, subatomic, chromo, and tachyon dynamics respectively corresponding to the scale β = g, p, h, f, e, c, m, a, s, k and t. As well, the associated non equilibrium scales are listed on the right hand side (RHS) in Fig The classical fluid mechanics covers the fields LFD, LED and LCD which are short for Laminar Fluid element Dynamics, Laminar Eddy Dynamics and Laminar Cluster Dynamics. In Fig. 2.2 four scales are selected to declare the relationship between the neighboring scales. As discussed in [3], each field described by a distribution function: f ( r, t, u ) β β β β (2.1) defines a system, which is composed of an ensemble of element, as well, each element is composed of an ensemble of small particles viewed as point mass atoms. For example, in Fig. 2.2 the element velocity of the smaller scale J becomes the atom velocity of the larger scale J+1: 5

24 Scale invariant Model of Statistical Mechanics UJ = u J+1 (2.2) and the system velocity in scale J becomes the element velocity in scale J+1: wj = U J+1 (2.3) Fig. 2.1 A scale invariant view of statistical mechanics from cosmic to tachyonic scales [3] 6

25 Chapter 2 Modified Equation of Fluid Motion FLUID ELEMENT EDDY CLUSTERS HYDRODYNAMIC SYSTEM FLUID ELEMENT L e λ e EDDIES w e = v h v e = u h u e = v c (J + 3/2) LED (J + 1) EED FLUID ELEMENT EDDY CLUSTERS L c EDDY CLUSTER MOLECULES λ c w c = v e v c = u e u c = v m (J + 1/2) LCD L c (J) ECD EDDY CLUSTER MOLECULES L m CLUSTER MOLECULE PARTICLES λ m w m = v c v m = u c u m = v a (J - 1/2) LMD (J - 1) EMD CLUSTER L m MOLECULE ATOMS L a L a λ a w a = v m v a = u m u a = v s (J - 3/2) LAD Fig. 2.2 Hierarchy of statistical fields for equilibrium eddy, cluster, and molecular dynamic scales and the associated non equilibrium fields [1] According to the scale invariant theory, flow parameters can be defined in a scale invariant way. As introduced in [3], the invariant definition of the density ρ β, and the relation between the velocities of element U (or local velocity), atom u and system w at neighboring scales are given as follows: ρ β = nm β β = mβ fd β u β (2.4) u β = U β-1 (2.5) U 1 β = ρ β mβ uβ fβdu β (2.6) wβ = U β+1 (2.7) 7

26 Convective, Diffusion and Local Velocities 2.2 Convective, Diffusion and Local Velocities At scale β, the system velocity w β can also be understood as mean velocity or convective velocity [18]: w β = U β (2.8) Following the classical theory [19 21], the fluid particle transfers inside a physical system due to two processes: diffusion and convection which is also known as advection. Therefore, using the scale invariant form, at the scale β, the local velocity U β can be expressed in terms of the convective velocity w β and the diffusion velocity v βgh : Uβ = wβ + v βgh (2.9) For most flow problems, the fluid diffusion contains two components: v = v + v βgh βg βh (2.10) In detail, v = D ln( ρ ) (2.11) βg β β which is related to the density dependent diffusion, and v = ν ln( U ) (2.12) βh β β which stands for the viscous influence to the diffusion. Incompressible flow assumption gives = 0 ρ β (2.13) Hence, in the scope of the present work, only the viscous component affects the diffusion: 8

27 Chapter 2 Modified Equation of Fluid Motion v = v = ν ln( U ) (2.14) βgh βh β β Substituting Eq. (2.14) into Eq. (2.9), one obtains Uβ = wβ + v βh (2.15) Eq. (2.15) plays a significant role in the derivation of the modified equation of fluid motion. In order to deliver a better understanding of this equation, a flow distribution arising from the source S is illustrated in Fig As shown in this figure, when Uβ = v βh (2.16) the fluid only performs the diffusion motion, the green region will be the final result of the fluid distribution. On the other hand, when U β = w β (2.17) the fluid only performs the convection motion, the resultant fluid distribution will be the yellow stream. However as mentioned above, in most cases the fluid particles transport with both diffusion and convection motions. In Fig. 2.3, this combination is the field inside the red ellipse. v gh W S U E w Fig. 2.3 Flow distribution near the source 9

28 Modified Equation of Fluid Motion for Incompressible Flow 2.3 Modified Equation of Fluid Motion for Incompressible Flow The modified equation of fluid motion is derived from the scale invariant form of the Boltzmann Equation [1]. The Boltzmann equation, often known as the Boltzmann transport equation can be used to study how a fluid transports the momentum, thus deriving transport properties, such as viscosity [22]. In this section, the derivation process of the modified equation of fluid motion will be described in detail. Furthermore, this modified equation will be compared with the Navier Stokes equation to highlight its potential advantages Derivation of the Modified Equation of Fluid Motion On the basis of the theory of non equilibrium statistical mechanics, Sohrab [1] gave the scale invariant form of the Boltzmann equation in the absence of the body force as follows: f β t δ fβ + u β fβ = (2.18) t Multiplying Avogadro Loschmidt number with an arbitrary invariant function of velocity ψ β and integrating them to both sides of Eq. (2.18), one obtains the Enskog equation of change in the scale invariant form: δ f β ( nβ ψ β ) + ( nβ ψ βu β ) = β d β t ψ u (2.19) δ t Defining ψ = m u β β β (2.20) in Eq. (2.19), one has δ fβ ( nm β βuβ) + ( nm β βuβuβ) = mβuβ du β (2.21) t δ t Note that the density definition in the statistical mechanics is ρ β = nm β β (2.22) 10

29 Chapter 2 Modified Equation of Fluid Motion Eq. (2.21) can be rewritten in the form: δ fβ ( ρβuβ) + ( ρβuβuβ) = mβ β d β t U U (2.23) δ t According to Sohrab [1], under the Stokes assumption, the term on the RHS of Eq. (2.23) results in 1 ( ρβ Uβ) + ( ρβuβuβ) = pβ + μ ( U β) (2.24) t 3 which shows the thermodynamic and hydrodynamic pressure. For incompressible flow, the hydrodynamic pressure can be neglected, therefore Eq. (2.23) is simplified as follows: ρβ β + ρβ ( β β) = pβ t U U U (2.25) Referring to Eq. (2.15), the second term in Eq. (2.25) can be expanded in the form: ρ ( U U ) = ρ [( w + v ) U ] β β β β β ρ ( U U β β βh ) = ρ ( w U ) + ρ ( v U ) β β β β β βh β β (2.26) Since the convective velocity is not a variable for the local flow field, Eq. (2.26) can be transferred as follows: ρ ( UU) = ρ w U + ρ ( v U ) (2.27) β β β β β β β βh β Substituting Eq. (2.12) into Eq. (2.27), the second term on the RHS of Eq. (2.27) becomes ρ ( v U ) = ρ {[ ν ln( U )] U } β βh β β β β 1 ρβ ( vβhuβ) = ρν β β [( Uβ) Uβ] U 2 ρ ( v U ) = ρν U β βh β β β β β β (2.28) 11

30 Modified Equation of Fluid Motion for Incompressible Flow Consequently, Eq. (2.27) becomes ρ ( UU) = ρ w U ρ ν U (2.29) 2 β β β β β β β β β As a result, Eq. (2.23) can be expressed as follows: p 2 β β + β β ν β β = ρβ t U w U U (2.30) which is known as the modified equation of fluid motion for incompressible flow. On the other hand, for compressible flow, this modified equation owns the similar expression. The detailed discussion can be found in the literature [1]. As presented in Eq. (2.30), the modified equation of fluid motion is a second partial differential equation. It is emphasized that the convective velocity w β is presented in the convection term of the equation. This introduced parameter can be calculated by Eq. (2.7) or Eq. (2.8). In other words, on one hand, for the current flow scale it is the system velocity as presented in Eq. (2.7), being equal to the local velocity at the next larger scale; on the other hand, the convective velocity can be understood as the mean velocity for the current scale system as shown in Eq. (2.8), which can be constructed theoretically for the simple flow model, e.g. the laminar boundary layer flow over a flat plate. In order to clarify these properties, numerical methods to determine the convective velocity will be discussed in Chapter 5. The relation between the convective velocities at two neighboring scales, the laminar flow scale and the potential flow scale is under consideration. For potential flow scale, the viscosity vanishes, so the diffusion velocity becomes v = ν ln( U ) = 0 (2.31) βh β β Continuing, U β gives U = w + v = w (2.32) β β βh β Therefore, for potential flow w β in Eq. (2.30) is replaced by U β and the scale invariant form of the Euler equation is obtained as follows: 12

31 Chapter 2 Modified Equation of Fluid Motion pβ β + β β = t U U U (2.33) ρ β This result indicates that, at the potential flow scale, the modified equation of fluid motion produces the same solution as the Euler equation. Therefore, according to Eq. (2.7) the solution for the convective velocity at the laminar flow scale can be constructed by using the velocity at the next larger scale, the potential flow scale. This relation can be expressed by the following equation: w = U (2.34) laminar potential For simplicity, the subscript symbol β will be neglected for the scale invariant form equations in the following discussions Relation between the Modified Equation and the Navier Stokes Equation The classical Navier Stokes equation for incompressible flow arises from applying Newton s second law to fluid motion, together with the assumption that the fluid stress is the sum of a diffusing viscous term, plus a pressure term [23]. For incompressible flow, the Navier Stokes equation can be expressed as: 2 p + ν = t U U U U (2.35) ρ with the consideration of the continuity equation U =0 (2.36) For small Knudsen number, Kn << 1, the derivation of the Navier Stokes equation for incompressible flow from the Boltzmann equation is also possible [24]. This recovers the connection between the Navier Stokes equation and the modified equation of fluid motion, that both of them can be applied to analyze transport phenomena. Restricting attention to the second term of Eq. (2.30) and Eq. (2.35) on the left hand side (LHS), it is obvious that the convective velocity w can replace the local velocity U. In mathematics, this important feature eliminates the non linearity. As a numerical investigation, the present research is mainly focused on the linear property of the modified equation of fluid motion. 13

32 Reynolds Averaged Modified Equation of Fluid Motion 2.4 Reynolds Averaged Modified Equation of Fluid Motion In CFD, it is very difficult for the numerical method to account the complete unsteady turbulence problems because of the complexity and significant computer time requirement. In most cases, the time averaged equation can deliver the net effect of the turbulence perturbation. Similar to the Reynolds averaged Navier Stokes equation (RANS), the time averaged form of the modified equation of fluid motion in conjunction with the isotropic turbulence viscosity hypothesis is considered for the further numerical applications to handle the turbulence problems. Turbulent eddies produce the fluctuation of velocity and pressure in both space and time. Given the mean velocity, U m, mean pressure p m and the corresponding turbulent fluctuation values, U and p, the transient expressions for velocity and pressure at the turbulence scale are U = U + U t tm t (2.37) pt = ptm+ p t (2.38) Arising from the quantum mechanics, Sohrab [25] described another understanding of the turbulence transition and presented that the convective velocity at the turbulence scale can be derived from the local velocity at the laminar scale as: w = U tm laminar (2.39) Potential Flow w potential = U w potential = U la min ar potential Laminar Flow w tm = U laminar Turbulent Flow Fig. 2.4 Flow chart of the convective velocity calculation process 14

33 Chapter 2 Modified Equation of Fluid Motion Therefore, concluding Eqs. (2.32), (2.34) and (2.39), the convective velocity for the turbulence scale is sourced from the potential flow scale, and developed in the laminar flow scale, as shown in the flow chart of Fig Following this turbulence transition analysis, the transient expression for the convective velocity w is w = w + w t tm t (2.40) According to the averaging method, the averaged value for the transient pressure and velocities are p = p fff p = 0 (2.41) t tm t U = U fff U t tm t = 0 (2.42) w = w fff w = 0 (2.43) t tm t Applying the results above, at the turbulence scale, the time averaged results for each component in Eq. (2.30) can be obtained as follows: U t t U = t tm (2.44) w U = ( w U ) t t t t w U t t = ( w U + w U ) tm tm t t w U = ( w U ) + ( w U ) (2.45) t t tm tm t t ( U ) = U 2 2 t tm (2.46) ( p t ) p = ρ ρ t t tm (2.47) 15

34 Reynolds Averaged Modified Equation of Fluid Motion In isotropic turbulence, the convective velocity shows the transient property of the system. Its transient fluctuation component should own the equivalent averaged value as that of the local velocity. In other words, one has wu = UU t t t t (2.48) Consequently, Eq. (2.45) becomes w U = ( w U ) + ( U U ) (2.49) t t tm tm t t Finally, the Reynolds averaged modified equation of fluid motion for turbulent flow is U t tm p + w U ν U = ( U U ) (2.50) 2 tm tm tm tm t t ρtm Eq. (2.50) represents the Reynolds averaged modified equation of fluid motion. In this equation, all time dependent terms are dropped since only steady states are considered. For incompressible flow, the Reynolds stress ρ ( UU t t ) has the same structure and dimension as the viscous stress tensor. However, instead of stress, it is just a re worked contribution of the fluctuating velocities to the change of the averaged ones. Same as the RANS, the Reynolds stress in the Reynolds averaged modified equation of fluid motion can be described by the turbulent kinetic energy (k) and the rate of energy dissipation (ε). This feature indicates that, the Reynolds stress in Eq. (2.50) can be handled by the turbulence model in numerical applications. Especially considering the bottleneck of the analytical method described in Chapter 1, the present analysis offers another opportunity for the modified equation of fluid motion to get the access to solve the turbulence problems. In addition, this will also simplify the coding of the numerical solver for modeling the turbulent flow. 16

35 Chapter 3 Numerical Error 3 Numerical Error In order to solve the modified equation of fluid motion for turbomachinery flow problems, a numerical solver is required. FVM is well suited for the numerical simulation of the Navier Stokes equation in the field fluid dynamics. Considering the relation between the modified and Navier Stokes equations, in this research, the FVM is used to establish the numerical solver for the modified equation. FVM is a discretisation method for representing and evaluating partial differential equations in the form of algebraic equations [26 27]. Inevitably, the discretisation process generates truncation and discretisation errors in the numerical solution. Before establishing the numerical solver, an estimation of these numerical errors derived from the discretisation of the modified equation of fluid motion is presented in this chapter. Both modified and Navier Stokes equations describe the transport phenomena of fluid dynamics. By starting from the generic transport equation, this chapter discusses the truncation and discretisation errors of the aforementioned two equations. Then the emphasis is placed on the differences of the numerical errors between these two equations to highlight the linear advantages of the modified equation in numerical applications. The transport equation for incompressible flow can be written in the differential form: φ + Ω φ ν φ = (3.1) t ρ S 2 φ where φ is the transport variable, ν is the diffusion coefficient, Ω stands for the local velocity U in the Navier Stokes equation, or the convective velocity w in the modified equation of fluid motion and S φ is the source that can either create or destroy φ. On the LHS of Eq. (3.1), the third component, which describes the diffusion property, contains the second derivative of φ in space, therefore Eq. (3.1) is a second order equation. As discussed by Patankar [28], the order of the discretisation has to be equal or higher than the order of Eq. (3.1) to deliver a satisfactory accuracy for the discretisation solution. 17

36 Taylor Series Expansion 3.1 Taylor Series Expansion W P E Δx Δx Fig D mesh Assuming the one dimensional (1D) grids with the uniform space interval Δx, as sketched in Fig. 3.1, the Taylor series expansion for the transport variable φ on this grid will be φ E φ x x x P x φ Δ φ Δ φ Δ = φ + Δ HOT x x 2! x 3! x 4! (3.2) φ W φ x x x P x φ Δ φ Δ φ Δ = φ Δ HOT x x 2! x 3! x 4! (3.3) where HOT is short for high order terms. Substracting Eq. (3.3) from Eq. (3.2) produces 3 3 φ φ Δx φe φw = 2 Δ x HOT 3 x x 3! (3.4) Dividing by 2Δx on both sides of Eq. (3.4), φ can be expressed as follows: 3 2 φ φe φw φ Δx = + HOT 3 x 2Δx x 3! (3.5) Setting O(Δx p ) as the truncation error in Taylor series, one obtains φ φe φw φ = = + O Δx x 2Δx 2 ( ) (3.6) where φ Δx x 3! O( Δ x ) = + HOT 3 (3.7) Now φ is expressed by a second order accurate central difference interpolation. 18

37 Chapter 3 Numerical Error If Δx is replaced by a time step size Δt, and points W, P, E are replaced by time steps (n-1) th, n th and (n+1) th, the second order accurate temporal derivative can be obtained as: n 1 n 1 φ φ + φ = + O Δt t 2Δt 2 ( ) (3.8) If the first order accurate discretisation solution (Eq. (3.2)) is applied to the temporal derivation, at point P, one has n+ 1 n φ φ φ P P P = + O( Δt) t Δt (3.9) which is well known as the forward method. On the other hand, 2 φ can be obtained by the summation of Eq. (3.2) and Eq. (3.3), as follows: φ Δx φ Δx φe + φw = 2φP HOT 2 4 x 2! x 4! (3.10) Dividing by Δx 2 on both sides of Eq. (3.10), φ can be explicitly expressed: φ φ 2φ + φ φ Δ = W P E 2 x + HOT x Δx x 4! (3.11) Setting φ Δx x 4! O( Δ x ) = 2 + HOT 4 (3.12) the second order accurate discretisation for 2 φ can be written in the form: 2 φw 2φP + φe 2 φ = + O( Δx ) 2 Δx (3.13) Referring to Eqs. (3.6), (3.9) and (3.13), finally the discretisation solution of the transport equation on the 1D mesh can be obtained as follows: φ n+ 1 n n n n n n P φp φe φw φw 2φ S P + φe φ + Ω ν = 2 Δt 2Δx Δx ρ (3.14) 19

38 Truncation Error where the time discretisation is first order and the space discretisation is second order. Since the source will not make influence to the discretisation difference between the modified and Navier Stokes equations, this component is not discussed in the present work. 3.2 Truncation Error Referring to Eqs. (3.2), (3.7) and (3.12) discussed in section 3.1, the truncation error of the transport equation (Eq. (3.1)) is φ Δ t φ x φ x + HOT + Ω ( Δ + HOT ) + ν (2 Δ + HOT ) t 2 x 3! x 4! Since the time step Δt can be well controlled during the numerical simulation, and for most fluid the kinematic viscosity ν is much smaller compared to the velocity Ω, the convection component in the transport equation produces the maximum truncation error. On the other hand, the Navier Stokes equation discretisation result differs from the modified equation of fluid motion only in the convection component. Consequently, it becomes more valuable to concentrate the analysis on this component. For Navier Stokes equation, one has Ω φ Δx φ Δx ( + HOT ) = ( + HOT ) 3 3 x 3! U x 3! (3.15) while, for the modified equation of fluid motion, one has Ω φ Δx φ Δx ( + HOT ) = ( + HOT ) 3 3 x 3! w x 3! (3.16) For both equations, the special interval Δx can be trended to zero, so that the truncation error trends to zero. However, in practice, one always has Δ x > 0 Hence, the truncation error is strongly dependent on Ω. During the numerical computation of the Navier Stokes equation, the local velocity U, experiences large oscillation and converges in the end of the computation. Consequently, the associated truncation error will be strongly influenced by this numerical oscillation. However, in the modified equation of fluid motion, the 20

39 Chapter 3 Numerical Error convective velocity is determined from another scale and preserved as a vector field with fixed values for the current scale, thus avoiding the aforementioned oscillation during the numerical computation process. Therefore, after the discretisation is carried out, although the modified equation cannot reduce the truncation error, independent of the accuracy order, it can produce a relative stable truncation error as compared to the Navier Stokes equation. Further, this advantage will make more influence to the accuracy of the numerical solution when the first order accurate discretisation method is applied, e. g. the upwind method. A numerical example about this will be discussed in section 5.1. On the other hand, this interesting result stimulates the comparison of the stability between these two equations. According to [28], the stability condition for the transport equation Eq. (3.1) is Ω 2 ν / Δx (3.17) which is also well known as the Peclet number. Referring to Pe 2, for Navier Stokes equation, one has U ν / Δx 2 (3.18) Since U is a variable varying during the numerical solution, one can only reduce Δx to preserve the stability of the equation. Hence a high quality mesh is required for the Navier Stokes equation discretisation to increase the stability of the associated numerical solution. While for the modified equation of fluid motion, one has w ν / Δx 2 (3.19) The convective velocity w has been determined for the whole computational domain. With the help of the fixed convective velocity, Δx is relaxed from the requirement of the stability of the discretisation. As compared to the Navier Stokes equation, a relative coarse mesh can also be accepted by the modified equation of fluid motion for the numerical discretisation. In order to support this analysis, numerical examples concerning the mesh issues will be discussed in sections 5.1 and 5.3 for two dimensional (2D) and three dimensional (3D) flow problems respectively. 21